An Operator Semigroup in Mathematical Genetics

The aim of this book is to motivate mathematicians with basic interests in fields such as functional analysis or operator semigroup theory and probability to turn attention to applied problems, which require such higher-order mathematical tools. In other words, we would like to show that a mathematical view “from above” on applied problems in population genetics leads not only to interesting “mathematical games” but also to important insights which may be of value to population genetics.

Genome is constituted by all heritable or genetic material coded in the DNA. In humans, it is organized in 23 pairs of chromosomes in each cell’s nucleus consisting of the total of twice \(3\times 10^{9}\) bases or symbols, as well as of thousands of copies of the relatively small circular mitochondrial genomes each consisting of about 16,600 symbols. Human nuclear genome is diploid, because the two sets of chromosomes are separately inherited from the two parents. The inheritance pattern follows Mendel’s Laws, whereby the sex cells called gametes contain one of the two sets of parental chromosomes, and offspring are formed by a fusion of two parental gametes, sperm and egg. In the neutral (no selection) case of the continuous-time Moran model, the time to the most recent common ancestor of a pair of individuals is exponentially distributed. We derive a differential equation (the “Master Equation”), which describes evolution of joint distributions of alleles in pairs of randomly sampled chromosomes. Analysis of this equation and its consequences is the principal topic of the book.

A special case of our master equation, specialized for microsatellite mutations, was derived to study influence of bottlenecks in modern human history. Jorde and co-workers analyzed allele frequency distributions at 60 tetranucleotide loci in a worldwide survey of human populations. Kimmel and co-workers investigated whether there is imbalance between allele size variances and heterozygosity observed in these data, as analyzed by an imbalance index they introduced. Three major groups of population, Asians, Africans, and Europeans, were considered for this purpose. The analysis shows that the data are consistent with a population bottleneck in modern humanity history and gradual settling of Europe and Asia.

We commence a systematic study of the master equation: In this chapter we introduce the background mathematical material needed for this study. To be more precise, this chapter presents basic ideas underlying functional-analytic approach to (countable state, continuous-time) Markov chains which, to recall, serve as a model of mutations. Starting with linear, normed and complete spaces, we discuss operators, and their convergence, to treat semigroups of operators related to Markov chains at the end of the chapter. Since these semigroups act in the space of absolutely summable sequences, which is a particularly simple Banach space, some results of the theory of semigroups of operators can be proved in a simpler way here.

We have found, under assumption of existence of the limit distribution of allelic states at pairs of chromosomes, explicit forms of the limit. Depending on the limit behavior of population size, we obtained different limit joint probability distributions of pairs. An interesting example of application of expression for the constant population size limit of the joint distribution is the model of microsatellite mutation with lower and upper bounds on the microsatellite size. From the viewpoint of microsatellite models, this is an unusual situation, since most of them, with a notable exception of Durretts’s model do not have a limit distribution of repeat count.

Population genetics is one of the constitutive parts of evolution. It provides mechanisms by which populations change given demography, environmental pressures and interactions with other populations, mutations and crossovers, not mentioning such exotic mechanisms as retro-transpositions, horizontal transfer and other. Population geneticists built great and logical machinery, which is propelled by abstract forces of drift, selection, mutation and recombination. The principles are simple, and encompass various versions of the Wright-Fisher and Moran Models of drift, with Markovian mutations, Haldane model for recombination and several types of selection. However, when combined, these principles lead to very complicated and frequently puzzling models some with very interesting mathematics.